is compact (see e.g. [1]}).
Consequently the mappings \(H^1(M)\ni u\mapsto e^u\in L^p(M)\)  are compact for all \(p \ge 1\) .
This will be crucial for the compactness of the problem, namely this compactness of Moser-Trudinger embedding essentially guarantees that validity of the Palais–Smale condition, as will be clear later.
